A192718 Elements of A192628 which are congruent to 7 (mod 8) (equivalently, 7 (mod 16)).

7, 55, 71, 87, 103, 119, 183, 263, 279, 343, 375, 391, 439, 455, 519, 551, 567, 583, 615, 631, 647, 695, 711, 727, 759, 775, 791, 823, 855, 871, 887, 903, 951, 967, 1015, 1047, 1079, 1095, 1111, 1127, 1159, 1175, 1191, 1223, 1239, 1271, 1303, 1319, 1367

Offset: 1

Author

Alexander Riasanovsky, Dec 31 2012

Keywords

• nonn

Comments

This is the subsequence/subset of A192628 which contains elements congruent to 7 modulo 8. Equivalently, these elements are also congruent to 7 modulo 16.

By partitioning A192628 into congruence classes k modulo 8, it turns out that it contains only elements congruent to 0, 1, 3, and 7 modulo 8. Further, the congruence classes 0, 1, and 3 modulo 8 are vanishing--having a density asymptotic to 0.

However, the 7 modulo 8 congruence classes appears to have nonzero density, conjectured 1/32. A current upper bound on its density (thus the entire density of A192628) is 1/16.

References

• J. Cooper and A. Riasanovsky, On the reciprocal of the binary generating function for the sum-of-divisors, Journal of Integer Sequences (accepted).
• J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.

Links

• J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8

Programs

Sage

prec = 2^12
R = PowerSeriesRing(GF(2), 'q', default_prec = prec)
q = R.gen()
sigma = lambda x : 1 if x == 0 else sum(Integer(x).divisors())
SigmaSeries = sum([sigma(m)*q^m for m in range(prec)])
SigmaBarSeries = 1/SigmaSeries
SigmaBarList = SigmaBarSeries.exponents()
SigmaBar7Mod8 = [m for m in SigmaBarList if mod(m, 8) == 7]
print(SigmaBar7Mod8)